For any \(\mathbf {x}\in \mathbb {C}^{N}\) , we can obtain \(\mathbf {x}^{H}(\mathbf {M}_{2}-\mathbf {M}_{1})\mathbf {x}\) 
\(=\mathbf {x}^{H}\mathbf {a}^{H}\mathbf {a}\mathbf {x}-\mathbf {x}^{H}\mathsf {diag}(\mathbf {a}^{H})\mathsf {diag}(\mathbf {a})\mathbf {x}=\sum _{i=1}^{N}\bigl |[\mathbf {a}]_{i}\bigr |^{2}\sum _{i=1}^{N}\bigl |x_{i}\bigr |^{2}-\sum _{i=1}^{N}\bigl |[\mathbf {a}]_{i}\bigr |^{2}\bigl |x_{i}\bigr |^{2}\)\(\ge 0\) ,
which completes the proof.

Lemma REF  shows the chosen minorizing function
of this paper is tighter than the widely used one in the literature,
where the largest eigenvalue scaled identity is used to majorize the
sum-of-rank-1 matrix [1]}, [2]}. This can
partially explain the faster convergence speed of the proposed algortihm.
